From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

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for internal DLA and related growth models

Amine Asselah, Alexandre Gaudilliere

To cite this version:

Amine Asselah, Alexandre Gaudilliere. From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models. Annals of Probability, Institute of Mathematical Statistics, 2013, 41 (3A), pp.1115-1159. �10.1214/12-AOP762�. �hal-00517593v3�

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Institute of Mathematical Statistics, 2013

FROM LOGARITHMIC TO SUBDIFFUSIVE POLYNOMIAL FLUCTUATIONS FOR INTERNAL DLA AND RELATED

GROWTH MODELS¹

By Amine Asselah and Alexandre Gaudilli`ere

Universit´e Paris-Est and Universit´e de Provence Dedicated to Joel Lebowitz, for his 80th birthday

We consider a cluster growth model onZ^d, called internal diffu- sion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimensiond≥2. In so doing, we intro- duce a closely related cluster growth model, that we callthe flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blach`ere in our Appendix) on the expected time spent by a random walk inside an annulus.

1. Introduction. The internal DLA cluster of volume N, say A(N), is obtained inductively as follows. Initially, we assume that the explored region is empty, that is, A(0) =∅. Then, consider N independent discrete-time random walks S₁, . . . , S_N starting from 0. For k≤N, assume A(k−1) is obtained, and define

τ_k= inf{t≥0 :S_k(t)∈/A(k−1)} and A(k) =A(k−1)∪ {S_k(τ_k)}.

Received May 2010; revised April 2012.

1Supported by GDRE 224 GREFI-MEFI, the French Ministry of Education through the ANR BLAN07-2184264 grant, by the European Research Council through the “Advanced Grant” PTRELSS 228032.

AMS 2000 subject classifications.60K35, 82B24, 60J45.

Key words and phrases.Internal diffusion limited aggregation, cluster growth, random walk, shape theorem, logarithmic fluctuations, subdiffusive fluctuations.

This is an electronic reprint of the original article published by the Institute of Mathematical StatisticsinThe Annals of Probability,

2013, Vol. 41, No. 3A, 1115–1159. This reprint differs from the original in pagination and typographic detail.

1

In such a particle system, we call explorers the particles. We say that thekth explorer is settled on S_k(τ_k) after time τ_k, and is unsettled before time τ_k. The clusterA(N) consists of the positions of the N settled explorers.

The mathematical model of internal DLA was introduced first in the chemical physics literature by Meakin and Deutch [13]. There are many industrial processes that look like internal DLA; see the nice review pa- per [7]. The most important seems to be electropolishing, defined as the improvement of surface finish of a metal effected by making it anodic in an appropriate solution. There are actually two distinct industrial processes (i) anodic leveling or smoothing which corresponds to the elimination of sur- face roughness of height larger than 1 micron, and (ii) anodic brightening which refers to elimination of surface defects which are protruding by less than 1 micron. The latter phenomenon requires an understanding of atom removal from a crystal lattice. It was noted in [13] that, at a qualitative level, the model produces smooth clusters, and the authors wrote, “it is also of some fundamental significance to know just how smooth a surface formed by diffusion limited processes may be.”

Diaconis and Fulton [2] introduced internal DLA in mathematics. They allowed explorers to start on distinct sites, and showed that the law of the cluster was invariant under permutation of the order in which explorers were launched. This invariance, namedthe abelian property, was central in their motivation. They treat, among other things, the special one-dimensional case.

In dimension two or more, Lawler, Bramson and Griffeath [10] prove that in order to cover, without holes, a sphere of radius n, we need about the number of sites of Z^d contained in this sphere. In other words, the asymp- totic shape of the cluster is a sphere. Then, Lawler in [9] shows subdiffusive fluctuations. The latter result is formulated in terms of inner and outer er- rors, which we now introduce with some notation. We denote withk · k the Euclidean norm onR^d. For any x inR^d and r inR, set

B(x, r) ={y∈R^d:ky−xk< r} and B(x, r) =B(x, r)∩Z^d. For Λ⊂Z^d, |Λ| denotes the number of sites in Λ. The inner errorδ_I(n) is such that

n−δ_I(n) = sup{r≥0 :B(0, r)⊂A(|B(0, n)|)}. Also, the outer errorδ_O(n) is such that

n+δ_O(n) = inf{r≥0 :A(|B(0, n)|)⊂B(0, r)}. The main result of [9] reads as follows.

Theorem 1.1 (Lawler). Assume d≥2. Then

P(∃n(ω) :∀n≥n(ω) δ_I(n)≤n^1/3log(n)²) = 1 (1.1)

and

P(∃n(ω) :∀n≥n(ω) δ_O(n)≤n^1/3log(n)⁴) = 1.

(1.2)

Since Lawler’s paper, published 15 years ago, no improvement of these estimates was achieved, but it is believed that fluctuations are on a much smaller scale than n^1/3. Moreover, (1.1) and (1.2) are almost sure upper bounds on errors, and no lower bound on the inner or outer error has been established. Computer simulations [3, 14] suggest indeed that fluctuations are logarithmic. In addition, Levine and Peres studied a deterministic ana- logue of internal DLA, the rotor-router model, introduced by Propp [6].

They bound, in [12], the inner error δ_I(n) by log(n), and the outer error δO(n) by n^1−1/d.

Our main result is the following improvement of Theorem 1.1.

Theorem 1.2. Assumed≥2. There is a positive constant A_d such that P(∃n(ω) :∀n≥n(ω) δ_I(n)≤A_dlog(n)) = 1

(1.3) and

P(∃n(ω) :∀n≥n(ω) δ_O(n)≤A_dlog²(n)) = 1.

(1.4)

Note added in proof. At about the same time, and with an independent approach, Jerison, Levine and Sheffield [5] obtained similar results with an improved bound on the outer error ind= 2. Then, by refining our approach, we obtained in [1] a bound of orderp

log(n) for both internal and external errors in dimension three or more. Jerison, Levine and Sheffield [4] did the same by following their approach.

Our approach builds on the work of Lawler, Bramson and Griffeath [10], which we review later. It also deals with more general models of diffusion limited aggregation which we now describe. Indeed, we introduce a fam- ily of cluster growth models for which a control of the fluctuations of the cluster shape is easily obtained. These growth models are built so that the asymptotic shape is spherical, but still they exhibit a large diversity of fluc- tuations parametrized by a certain width ranging from a large constant to a power 1/3 of the radius of the asymptotic sphere. Moreover, all these clusters are coupled to internal DLA, and, as a consequence, we obtain logarithmic bounds on the fluctuations for internal DLA. We generalize internal DLA by allowing explorers to settle only at some special times. Thus, each explorer iis associated with a collection of times{σ_i,k, k∈N} and

τ_i^∗= inf{σ_i,k:S_i(σ_i,k)∈/A^∗(i−1)} and A^∗(i) =A^∗(i−1)∪ {S_i(τ_i^∗)}. The internal DLA is recovered as we chooseσ_i,k=kfor all i= 1, . . . , N and k∈N. We call{σ_i,k, k∈N}theflashing times associated to theith explorer, and {S_i(σ_i,k), k∈N} its flashing positions.

Fig. 1. Cell decomposition, and flashing positions as stars.

In this paper, we consider stopping times of a special form, linked with the spherical nature of the internal DLA cluster. An illustration with one flashing explorer’s trajectory is made in Figure 1.

The precise definition of the flashing times requires additional notation, which we postpone to Section 3. We describe here key features of flashing processes. We first choose a sequence of widths, say H={h_n, n∈N}, and then partitionZ^dinto concentric shells{Sn, n∈N}, whose respective widths are{2h_n, n∈N}. Each shell is in turn partitioned into cells, which are brick- like domain, of side length equal to the width of the shell. The flashing times are chosen such that (i) an explorer flashes at most once in each shell, (ii) the flashing position, in a shell, is essentially uniform over the cell an explorer first hits upon entering the shell and (iii) when an explorer leaves a shell, it cannot afterward flash in it.

For a given sequenceH, we call the process just described the H-flashing process. Note that feature (ii) is the seed of a deep difference with internal DLA. The mechanism of covering a cell, for the flashing process, is very much the same as completing an album in the classical coupon-collector pro- cess. Thus, we need of the order of Vlog(V) explorers to cover a cell of volume V. For internal DLA, with explorers started at the origin, we only need of order V explorers to cover a sphere of volume V as shown in [10], and we believe that we need a number of explorers of order |C| to cover a cell C, even if they start on the boundary of the cell. In addition, feature (ii) allows us tolocalize the covering mechanism, in the sense that a particle entering a shell cannot flash outside the cell through which it entered that shell. Finally, feature (iii) is essential for having a useful coupling between flashing and internal DLA processes.

Lemma 1.3. Assume that N is an integer, and His a sequence of posi- tive integers. There is a coupling between the two processes, using the same trajectories S₁, . . . , S_N such that

A(N) =

N

[

i=1

{S_i(T(i))} and A^∗(N) =

N

[

i=1

{S_i(T^∗(i))} (1.5)

and T^∗(i)≥T(i) for alli= 1, . . . , N.

As a corollary of Lemma1.3, we have the following useful result.

Corollary 1.4. Under the hypotheses of the previous lemma, fork≥1:

• if A^∗(N)⊂S

j<kSj, then A(N)⊂S

j<kSj;

• if S

j<kSj⊂A^∗(N), then S

j<kSj⊂A(N).

An H-flashing process, with h_j≥h₀ for j≥0, and h₀ a large constant, produces a clusterA^∗(N), for which we bound easily the inner error,δ^∗_I(n).

Then, to bound the outer error,δ^∗_O(n), we follow the approach of [9], though with a slightly simpler proof.

Proposition 1.5. Assume that for j≥1, h_j≤h_j+1≤(1 +_2j¹)h_j, with a large h₀. For a positive constant A^∗_d, we have

P(∃n(ω) :∀n≥n(ω) δ^∗_I(n)≤A^∗_dh(n) log(n)) = 1 (1.6)

and

P(∃n(ω) :∀n≥n(ω) δ_O^∗(n)≤A^∗_dh(n) log²(n)) = 1, (1.7)

where h(n) = max{h_k∈R:r_k≤n}.

Finally, we establish lower bound on the inner and outer error.

Proposition 1.6. Assume that h₀ is large enough. Then, there is a constant a^∗_d such that

P(∃n(ω) :∀n≥n(ω) δ_I^∗(n)≥a^∗_dh(n) log(h(n))) = 1 (1.8)

and

P(∃n(ω) :∀n≥n(ω) δ^∗_O(n)≥a^∗_dh(n) log(n)) = 1.

(1.9)

Corollary 1.4 and Proposition 1.5, with the choice h_j =h₀ for all j >0, imply Theorem1.2which deals with internal DLA.

Let us now review previous work on internal DLA.

On previous bounds for internal DLA. We describe the approach of [10], for establishing the upper bound for the inner error. It is convenient to consider explorers starting outside the origin with initial configuration de- noted η. We denote also by A(Λ, η) the cluster obtained from explorers initially onη, with an explored region Λ⊂Z^d.

Now, for a site z∈Z^d, we call W(η, z) [resp., M(η, z)] the number of explorers (resp., of random walks) which visitz before settling. For an inte- gern, andηconsisting of|B(0, n)|explorers at the origin, the authors of [10]

first write

{B(0, r)6⊂A(∅, η)} ⊂ [

z∈B(0,r)

{W(η, z) = 0}.

Then, they look for the largest value of r_n (in terms of n) which guaran- tees that|B(0, r_n)| ×sup_z∈B(0,rn)P(W(η, z) = 0) be the term of a convergent series.

The approach of [10] is based on the following observations. (i) If explorers would not settle, they would just be independent random walks; (ii) exactly one explorer occupies each site of the cluster. Thus, the following equality holds in law:

W(η, z) +M(A(∅, η), z)≥M(η, z).

Now, an observation of Diaconis and Fulton [2] is that we can realize the cluster by sending manyexploration waves. Let us illustrate this observation with two waves. We first stop the explorers on the external boundary of a ball of radiusR, say ∂B(0, R). The cluster consisting of the positions of settled explorers is denotedA_R(∅, η), so thatA_R(∅, η)⊂B(0, R). The configuration with stopped explorers on∂B(0, R) is denotedζR(η). Then, the second wave consists in launching the explorers of ζ_R(η), with explored regionA_R(∅, η).

In other words, we have an equality in law

A(∅, η) =A_R(∅, η)∪A(A_R(∅, η), ζ_R(η)).

Moreover, if the indexR refers only to explorers (or walks) of the first wave, then forz∈B(0, R),

W_R(η, z) +M_R(A_R(∅, η), z)≥M_R(η, z).

(1.10)

The authors of [10] considerR=nandz∈B(0, n). SinceW(η, z)≥W_n(η, z), we have using (1.10), for anyα >0,

P(W(η, z) = 0)≤P(M_n(η, z)< α) +P(M_n(B(0, n), z)> α).

(1.11)

We then look for siteszsuch thatE[M_n(η, z)]> α > E[M_n(B(0, n), z)] (and η=|B(0, n)|δ₀). Note thatM_n(η, z) andM_n(B(0, n), z) are sums of indepen- dent Bernoulli variables with well-known large deviation estimates. If we set

2α=E[M_n(η, z)] +E[M_n(B(0, n), z)]

and

˜

µ_n(z) =E[M_n(η, z)]−E[M_n(B(0, n), z)], then

P(Mn(η, z)< α)≤exp

−(E[M_n(η, z)]−α)² 2E[M_n(η, z)]

(1.12)

≤exp

− µ˜²_n(z) 8E[M_n(η, z)]

.

Lawler in [9] establishes that for z∈B(0, n),

E[M_n(η, z)]∼n(n− kzk) and µ˜_n(z)∼(n− kzk)².

Replacing these values in (1.12), the bound n− kzk ≥n^1/3log(n) is such thatP(W_n(η, z) = 0) is the term of a convergent series.

We now sketch our main ideas leading to logarithmic fluctuations for internal DLA.

On logarithmic fluctuations. Our approach is inspired by Lawler, Bram- son and Griffeath’s work [10]. We develop three original ideas: (i) we propose a cluster growth model, the flashing process, whose covering mechanism is simpler than internal DLA; (ii) we look at an intermediary scale,the scale of cells, since the deviations of the number of visits decrease with the cell- length; (iii) we build a coupling between flashing process and internal DLA which allows us to transport bounds from one model to the other.

Let us describe how the idea of an intermediary scale is used in the context of flashing processes. Recall that we first partition Z^d into a sequence of concentric shells. Each shell is partitioned into cells whose side length equals the width of the shell. Now, we observe that a site has good chances to lie inside the cluster if some cell, say C, about this site, is crossed by many explorers. The notationW(η,C) refers to the number of explorers visitingC, when their initial configuration is η. We drop the index n appearing in W_n(η, z) since there are no more constraints on not escaping the ballB(0, n).

Now, the coupon-collector nature of the covering mechanism suggests that for some positive constantα_d,

W(η,C)≥α_d|C| ×log(|C|) (1.13)

=⇒ C ⊂A(∅, η) with a large probability.

We neglect in these heuristics the log(|C|) term in (1.13).

Note that in [10], all the explorers start from the origin, whereas here, we only know that theycross C. For internal DLA, estimating the probability thatC is not covered, whenC is large andW(η,C)≥α_d|C|raises a difficulty which is absent when considering flashing processes.

We now make our argument more precise. For a scale h and an integer K >1, to be determined, assume that B(0, n−Kh) is covered by settled

explorers. Partition the shell S=B(0, n−(K −1)h)\B(0, n−Kh) into about (n/h)^d−1 cells, each of volume h^d. It is also convenient to stop the explorers as they reach the boundary of B(0, n−Kh). Thus, with such a stopped process, explorers are either settled insideB(0, n−Kh) or unsettled but stopped on its boundary, denoted∂B(0, n−Kh). What we have called earlier the number of explorers crossing C is taken here to be the unsettled explorers stopped on C ∩∂B(0, n−Kh).

Assuming (1.13) holds, it remains to show that the probability of the event {∃C ∈ S:W(η,C)< α_d|C|} is small. We improve (1.11) by first using the independence between W(η,C) and M(B(0, n−Kh),C), and then by replacing A_R(∅, η) by B(0, n−Kh) in (1.10) with R=n−Kh and η=

|B(0, n)|10,

W(η,C) +M(B(0, n−Kh),C)≥M(η,C).

(1.14)

Also, we define

µ(C) =E[M(η,C)]−E[M(B(0, n−Kh),C)].

Now, using that M(η, z) and M(B(0, n−Kh), z)) are sums of independent Bernoulli variables, we show that (1.14) implies a Gaussian-type lower tail

P(W(η,C)< α_d|C|)≤exp

−(µ(C)−α_d|C|)² cν(C)

(1.15)

for a positive constantc, and where ν(C)

ν(C) = var(M(η,C))−var(M(B(0, n−Kh),C)).

We then show that bothµ(C) andν(C) are of orderK|C|. Then,P(W(η,C)<

α_d|C|) is summable as soon as K|C| ≥Alog(n).

Outline of the paper. The rest of the paper is organized as follows. Sec- tion 2 introduces the main notation, and recalls known useful facts. In Section 3, we build the flashing process, give an alternative construction through exploration waves and sketch the proof of Lemma1.3. In Section4, we prove Propositions 1.5 and 1.6 using the construction in terms of ex- ploration waves. In Section 5, we obtain a sharp estimate on the expected number of explorers crossing a given cell, and prove feature (ii) of the flash- ing times. Both proofs are based on classical potential theory estimates.

Finally, in theAppendix, we give a proof of Lemma1.3, and recall a result of S´ebastien Blach`ere.

2. Notation and useful tools.

2.1. Notation. We say that z, z^′∈Z^d are nearest neighbors when kz− z^′k= 1, and we writez∼z^′. For any subset Λ⊂Z^d, we define

∂Λ ={z∈Z^d\Λ :∃z^′∈Λ, z^′∼z}.

For anyr≤R, we define the annulus

A(r, R) =B(0, R)\B(0, r) and A(r, R) =A(r, R)∩Z^d. (2.1)

A trajectoryS is a discrete nearest-neighbor path onZ^d. That is,S:N→Z^d withS(t)∼S(t+ 1) for all integer t. For a subset Λ inZ^d, and a trajectory S, we define the hitting time of Λ as

H(Λ;S) = min{t≥0 :S(t)∈Λ}.

We often omit S in the notation when no confusion is possible. We use the shorthand notation

Bn=B(0, n), B_n=B(0, n), HR=H(B^c_R) and Hz=H({z}).

For anya,binRwe writea∧b= min{a, b}, anda∨b= max{a, b}. Let Γ be a finite collection of trajectories on Z^d. For R >0, z in Z^d and Λ a subset ofZ^d, we callM(Γ, R, z) [resp.,M(Γ, R,Λ)] the number of trajectories which exitB(0, R) on z (resp., in Λ).

M(Γ, R, z) =X

S∈Γ

1_{S(HR)=z} and M(Γ, R,Λ) =X

z∈Λ

M(Γ, R, z).

When we deal with a collection of independent random trajectories, we rather specify its initial configuration η∈N^Zd, so that M(η, R, z) is the number of random walks starting from η and hitting B(0, R)^c on z. Two types of initial configurations are important here: (i) the configurationn1z^∗

formed by n walkers starting on a given site z^∗ and (ii) for Λ⊂Z^d, the configuration1Λ that we simply identify with Λ. For any configuration η∈ N^Zd we write

|η|= X

z∈Z^d

η(z).

For any Λ⊂Z^d, we define Green’s function restricted to Λ, G_Λ, as follows.

For x, y∈Λ, the expectation with respect to the law of the simple random walk started at x, is denoted with E_x (the law is denoted P_x) and

G_Λ(x, y) =E_x

X

0≤n<H(Λ^c)

1_{S(n)=y}

.

In dimension 3 or more, Green’s function on the whole space is well defined and denotedG. That is, for any x, y∈Z^d,

G(x, y) =E_x

X

n≥0

1_{S(n)=y}

.

In dimension 2, the potential kernel plays the role of Green’s function a(x, y) = lim

n→∞E_x

" _n X

l=0

(1{S(l) =x} −1{S(l) =y})

# .

2.2. Some useful tools. We recall here some well-known facts. Some of them are proved for the reader’s convenience. This section can be skipped at a first reading.

In [10], the authors emphasized the fact that the spherical limiting shape of internal DLA was intimately linked to strong isotropy properties of Green’s function. This isotropy is expressed by the following asymptotics (Theo- rem 4.3.1 of [11]). In d≥3, there is a constantK_g, such that for any z6= 0,

G(0, z)− C_d kzk^d−2

≤ K_g

kzk^d withC_d= 2 v_d(d−2), (2.2)

where v_d stands for the volume of the Euclidean unit ball in R^d. The first order expansion (2.2) is proved in [11] for general symmetric walks with finite d+ 3 moments and vanishing third moment. All the estimates we use are eventually based on (2.2), and we emphasize the fact that the estimate is uniform in kzk. There is a similar expansion for the potential kernel.

Theorem 4.4.4 of [11] establishes that forz6= 0 (with γ the Euler constant),

a(0, z)− 2

πlog(kzk)−2γ+ log(8) π

≤ K_g

kzk². (2.3)

We recall a rough but useful result about the exit site distribution from a sphere. This is Lemma 1.7.4 of [8].

Lemma 2.1. There are two positive constants c₁, c₂ such that for any z∈∂B(0, n), and n >0

c₁

n^d−1 ≤P₀(S(H_n) =z)≤ c₂ n^d−1. (2.4)

We now state an elementary lemma.

Lemma 2.2. Each z^∗ in Z^d\ {0} has a nearest-neighbor z (i.e., z^∗∼z) such that

kzk ≤ kz^∗k − 1 2√

d. (2.5)

Proof. Without loss of generality we can assume that all the coor- dinates of z^∗ are nonnegative. Let us denote by b the maximum of these coordinates, and note that

kz^∗k²≤db² and b≥1.

(2.6)

Denote by z the nearest-neighbor obtained from z^∗ by decreasing by one unit a maximum coordinate. Using (2.6),

kz^∗k²− kzk²=b²−(b−1)²= 2b−1≥b≥kz^∗k

√d . (2.7)

Note that (2.5) follows from 2kz^∗k(kz^∗k−kzk)≥ kz^∗k²−kzk², and (2.7).

We state now a handy estimate dealing with sums of independent Bernoulli variables.

Lemma 2.3. Let{X_n, Y_n, n∈N}be independent 0–1 Bernoulli variables.

For integers n, m let S=X₁+· · ·+X_n and S^′=Y₁+· · ·+Y_m. Define for t∈R

f(t) =e^t−1−t and g(t) = (e^t−1)². If 0≤t≤log(2), then

E[exp(t(S−E[S]))]

E[exp(t(S^′−E[S^′]))]≤exp f(t)E[S−S^′] +g(t)

m

X

i=1

E[Y_i]²

! . (2.8)

Assume now that for κ >1, sup_nE[Y_n]≤^κ−_κ¹. If t≤0, then E[exp(t(S−E[S]))]

E[exp(t(S^′−E[S^′]))]≤exp f(t)E[S−S^′] +κ 2g(t)

m

X

i=1

E[Y_i]²

! . (2.9)

Proof. LetXbe a Bernoulli variable, andp=E[X]. Using the inequal- itye^x≥1 +x forx∈R, we have

E[exp(t(X−E[X]))] =pe^t(1−p)+ (1−p)e^−tp

=e^−pt(1 +p(e^t−1)) (2.10)

≤exp(f(t)E[X]).

For a lower bound, we distinguish two cases.

First, assume t≥0. We claim that exp(x−x²)≤1 +x for 0≤x≤1.

Indeed, we use three obvious inequalities:e^x≥1 +xforx∈R, (i) for x≤1, 1 +x+x²≥e^x, and (ii) (1 +x²)(1 +x)≥1 +x+x². Thus

e^x2(1 +x)≥(1 +x²)(1 +x)≥1 +x+x²≥e^x.

This yields the claim. Now, setx=p(e^t−1), so thatx≤1 whene^t≤2. The last inequality in (2.10) yields

E[exp(t(X−E[X]))]≥exp(−tp+p(e^t−1)−p²(e^t−1)²) (2.11)

=e^{f(t)p−g(t)p2}.

Assume now thatt≤0, and forκ >1,p < ^κ−1_κ . We claim that for 0≤x≤^κ−1_κ , exp

−x−κ 2x²

≤1−x.

(2.12)

Indeed, we have an additional inequality (iii) 1−x+^x₂² ≥exp(−x) when x≥0. Note also that

1 +κ

2x²

(1−x)≥1−x+x²

2 ⇐⇒ x≤κ−1 κ .

Thus

e^κx2/2(1−x)≥

1 +κ 2x²

(1−x)≥1−x+x²

2 ≥e^−x. Now, setx=−p(e^t−1)≥0, so that x≤^κ−_κ¹. We obtain

E[exp(t(X−E[X]))]≥exp

−tp+p(e^t−1)−κ

2p²(e^t−1)²

(2.13)

=ef(t)p−κg(t)p²/2.

Inequalities (2.8) and (2.9) follow (2.11) and (2.13).

3. The flashing process. In this section, we construct the flashing pro- cess, and state the crucial “uniform hitting property.” We then present a useful equivalent construction in terms of exploration waves. Finally, we explain the coupling of Lemma1.3, but postpone its proof to theAppendix.

3.1. Construction of the process.

Partitioning the lattice. We are given a sequence H={hn, n∈N}. We partition the lattice into shells (Sj:j≥0). For an illustration, see Figure 1.

For a given parameterh₀>0, the first shellS0is the ballB(0, h₀). Forj≥1, shell j is the annulus [see its definition (2.1)]

Sj =A(rj−hj, rj+hj),

where{r_j, j≥1} is defined inductively by r₁=h₀+h₁, and for j≥1, r_j+1−h_j+1=r_j+h_j.

In Section 4, we need that (o) H is increasing, (i) j7→h_j/r_j is decreasing and (ii) h_j =O(r_j^1/3). These properties are a straightforward consequence of our hypothesis hj≤hj+1≤(1 +_2j¹)hj. Actually we will only need these properties, and our hypothesis is no more than a sufficient condition.

We also define

Σ₀={0} and Σ_j=∂B(0, r_j), j≥1.

Flashing times. The key feature we expect from the flashing process is that its covering mechanism be simple. More precisely, our construction is guided by property (ii) of the Introduction which states that the flashing position,in a shell,is essentially uniform over the cell an explorer first hits upon entering the shell. Thus, we need to define together cells and flashing times to realize property (ii). It is important that all sites of a shell can be chosen as flashing sites with about the same frequency. In this respect, let us remark that a cell in shellSj cannot be a ball of radiush_j centered on Σ_j.

Indeed, if this were the case, sites at a distance abouth_j would be in much fewer cells than sites of Σ_j, and this would fail to make the covering of a shell uniform. We find it convenient to build a cell with a mixture of balls and annuli. A (random) flagY_j tells the explorers whether it flashes upon exiting either a sphere or the boundary of an annulus, whose distance from Σj is governed with a random radius R_j of appropriate density. Also, to allow for the possibility of flashing on its hitting position on Σj, we introduce an additional flagX_j.

More precisely, consider{Xj, Yj, j≥0}a sequence of independent Bernoulli variables such that

P(X_j = 1) = 1−P(X_j= 0) = 1 h^d_j and

P(Y_j= 1) = 1−P(Y_j= 0) =

1, if j= 0,

1

2, if j≥1.

Consider also a sequence of continuous independent variables {R_j, j≥0} each of which has densityg_j: [0, h_j]→R⁺ with

gj(h) =dh^d−1 h^d_j . (3.1)

For j≥0, and z_j in Σ_j, let S be a random walk starting inz_j, an define a stopping timeσ as follows. If R_j=hfor some h≤h_j, then

σ=







0, if X_j= 1,

H(B(z_j, h∧(r_j+h_j− kz_jk))^c), if X_j= 0 and Y_j= 1, H(A(r_j−h, r_j+h)^c), if X_j= 0 and Y_j= 0.

We set H_j=H(Σ_j), and we define the stopping times (σ_j:j≥0) as σj=Hj+σ(S◦θHj),

whereθstands for the usual time-shift operator. For j≥0 we note that, by construction, S(t)∈ Sj for all t such that H_j≤t < σ_j and we say that σ_j is a flashing time when S(σ_j) is contained in the intersection between Sj

and the cone with base B(S(H_j), h_j/2). We call such an intersection a cell centered at S(H_j), that we denoteC(S(H_j)). In other words, for anyz∈Σ_j

C(z) =Sj∩ {x∈R^d:∃λ≥0,∃y∈B(z, hj/2), x=λy}. (3.2)

The uniform hitting property. The main property of the hitting time σ constructed above is the following proposition, which yields property (ii) of the flashing process to be defined soon.

Proposition 3.1. There are two positive constants α₁< α₂, such that, for h0 large enough, j≥0, zj∈Σj, and z^∗∈ C(zj).

α₁ h^d_j ≤P_z

j(S(σ) =z^∗)≤α₂ h^d_j. (3.3)

The proof of Proposition3.1is given in Section 5.

The flashing process. Consider a family ofN independent random walks (S_i: 1≤i≤N) with their stopping times (H_i,j, σ_i,j:j≥0). Let also z_i,j= Si(Hi,j) be the first hitting position of Si on Σj.

We define the cluster inductively. Set A^∗(0) =∅. For i≥1, we define τ_i^∗ as the first flashing time associated withS_i when the explorer stands outside A^∗(i−1). In other words,

τ_i^∗= min{σ_i,j:j≥0, S_i(σ_i,j)∈ C(z_i,j)∩A^∗(i−1)^c} and

A^∗(i) =A^∗(i−1)∪ {S_i(τ_i^∗)}.

3.2. Exploration waves. Rather than buildingA^∗(N) following the whole journey of one explorer after another, we can buildA^∗(N) as an increasing union of clusters formed by stopping explorers on successive shells. Similar wave constructions are introduced in [10] and [9]. We use this alternative construction in the proof of Propositions1.5and 1.6.

We denote byξ_k∈(Z^d)^N the explorers positions after the kth wave. We denote by A^∗_k(N) the set of sites where settled explorers are after the kth wave. Our inductive construction will be such that

ξ_k(i)∈/Σ_k ⇔ ξ_k(i)∈ [

j<k

Sj ⇔ ξ_k(i)∈ A^∗k(N).

For k= 0 we set ξ₀(i) = 0, and A^∗0(i) =∅, for 1≤i≤N. Assume that for k≥0, A^∗_k(i) is built for i= 0, . . . , N. We set A^∗_k+1(0) =A^∗_k(N). For i in {1, . . . , N}, we set the following:

• If ξ_k(i)∈/Σ_k, then

ξ_k+1(i) =ξ_k(i)∈ [

j<k

Sj and A^∗k+1(i) =A^∗k+1(i−1).

• If ξ_k(i)∈Σ_k and S_i(σ_i,k)∈ C(z_i,k)∩ A^∗_k(i−1)^c, then

ξ_k+1(i) =S_i(σ_i,k)∈ Sk and A^∗k+1(i) =A^∗k+1(i−1)∪ {S_i(σ_i,k)}.

• If ξ_k(i)∈Σ_k and S_i(σ_i,k)∈ C/ (z_i,k)∩ A^∗_k(i−1)^c, then

ξ_k+1(i) =S_i(H_i,k+1)∈Σ_k+1 and A^∗k+1(i) =A^∗k+1(i−1).

In words, for each k≥1, during the kth wave of exploration, the unsettled explorers move one after the other in the order of their labels until either settling inSk−1, or reaching Σ_k where they stop. We then defineA^∗(N) by

A^∗(N) = [

k≥1

A^∗k(N).

We explain now why this construction yields the same cluster as our previous definition. An explorer cannot settle inside a shell it has left, and thus cannot settle in any shell Sj with j < k if it reaches Σ_k. Now, since each wave of exploration is organized according to the label ordering, the fact that an explorer has to wait for the following explorers before proceeding its journey beyond Σ_k does not interfere with the site where it eventually settles.

3.3. Coupling internal DLA and flashing processes.

Proof of Lemma1.3. For each positive integerN, we build a coupling be- tweenA(N) andA^∗(N). We first describe the main features of our coupling in words. Its precise definition is postponed to theAppendix.

We launch N independent random walks, and build inductively the asso- ciated clusters A(1), A(2), . . . , A(N). In doing so, we use the increments of these random walks to define, step by step,Nflashing trajectoriesS₁^∗, . . . , S_N^∗ up to some times ¯t₁, . . . ,t¯_N. Let us describe informally stepi+ 1 of the induc- tion. Assume thatS₁^∗, . . . , S_i^∗ are defined up to some timest1≤t¯1, . . . , ti≤¯ti, and that each site of A(i) is covered by exactly one S_k^∗(t_k) with 1≤k≤i.

We can think of S₁^∗(t₁), . . . , S_i^∗(t_i) as the positions of stopped flashing ex- plorers, some of them stopped at one of their flashing times—say on blue sites—some of them not—say onred sites. Then, we add thei+ 1th explorer and flashing explorer. We setS_i+1^∗ (0) =S_i+1(0) = 0. We add new increments both to S_i+1 and to the trajectory of one flashing explorer, say with label j in {1;. . .;i+ 1}, in such a way that the current position of the walker i+ 1 and that of the flashing explorer j coincide. The label j is defined inductively as follows. Initially,j=i+ 1. Assume now that the walker i+ 1 flashes on a red or blue site insideA(i). This site is occupied by exactly two stopped flashing explorers,j andj^′ [and all other red and blue sites ofA(i) are occupied by exactly one flashing explorer]. Since flashing explorers can settle at their flashing times, it makes sense, when j is flashing, to add the next increment to the trajectory of flashing explorer j^′ rather than j. We do so in two cases, first, when this happens on a red site. In this case, we turn blue that site since j is stopped at a flashing time. Second, when this happens on a blue site, say z, and j^′ > j. Note that in this case, both ex- plorers flash onz, but explorerj reachesz before explorerj^′ when launched in their label order. Our choice is such that the eventual clusterA^∗(N) has the correct law. In all other cases, we keep adding the increments of S_i+1

to the same flashing trajectory. It is important to note that the value of the increment does not depend on the index of the trajectory we choose to extend. Walkeri+ 1 eventually steps outsideA(i), say onz^∗, while following a flashing trajectory, say the jth one. We stop the jth flashing trajectory onz^∗, and paintz^∗ blue or red according to whetherz^∗ is one of its flashing sites or not.

When the last walker steps outside A(N−1), we have A(N) ={S^∗₁(¯t₁);. . .;S_N^∗ (¯t_N)} with|A(N)|=N.

(3.4)

To defineA^∗(N) we launch again, in their label’s order, the flashing explorers from their current positions (possibly some or none of them since some or all of them can already have reached their settling position). We then get

A^∗(N) ={S₁^∗(τ₁^∗);. . .;S_N^∗(τ_N^∗)} (3.5)

with |A^∗(N)|=N andτ_k^∗≥¯t_k for all k.

Proof of Corollary 1.4. Since a flashing explorer that visited some site beyond a given shell cannot settle in that shell, the one-to-one map

ψ_N:S_k^∗(¯t_k)∈A(N)7→S_k^∗(τ_k^∗)∈A^∗(N), k= 1, . . . , N, (3.6)

satisfies, for allk and l, S_k^∗(¯t_k)∈/ [

m<l

Sm ⇒ S_k^∗(τ_k^∗) =ψ_N(S_k^∗(¯t_k))∈/ [

m<l

Sm. (3.7)

Thus, for allN≥0 there is a coupling and a one-to-one mapψ_N between A(N) and A^∗(N) such that for all k≥1,

ψ_N(A(N)∩B^c

rk+hk)⊂A^∗(N)∩B^c

rk+hk. (3.8)

Inclusion (3.8) has two important consequences:

(a) If A^∗(N)⊂B_r

k+hk, then A(N)⊂B_r

k+hk. Indeed, any site in A(N) outsideB_r

k+hk produces, through ψ_N, a site in A^∗(N) outside B_r

k+hk. (b) IfB_r

k+hk⊂A^∗(N), thenB_r

k+hk⊂A(N). Indeed, those sites inA(N) that are mapped through ψ_N on A^∗(N)∩B_r

k+hk =B_r

k+hk are necessarily contained in B_r

k+hk. Since their number is |B_r

k+hk| and ψN is one-to-one, they completely coverB_r

k+hk.

4. Fluctuations. In this section, we prove Propositions 1.5 and 1.6. To do so we use the construction in terms of exploration waves of Section3.2.

Thus, we think of the growing cluster as evolving in discrete time, where time counts the number of exploration waves. The proofs in this section rely on potential theory estimates which we have gathered in Section 5, for the ease of reading.

4.1. Tiles. We recall that we have defined a cell of Sj in (3.2), as the intersection of a cone withSj. We need also a smaller shape. We define, for

any z_j in Σ_j, and for a small ε₀ to be defined later,

C˜(zj) =Sj∩ {x∈R^d:∃λ≥0,∃y∈B(zj, ε0hj), x=λy}. (4.1)

As in Lemma 12 in [9], concerning locally finite coverings, we claim that, for h₀ large enough, there exist a positive constant K_F, and, for each j≥0, a subset ˜Σ_j of Σ_j such that

∀y∈ Sj |{z∈Σ˜_j:y∈C˜(z)}| ≤K_F and Sj= [

zj∈Σ˜j

C˜(z_j).

(4.2)

For any z_j ∈Σ˜_j, we call tile centered at z_j, the intersections of ˜C(z_j) with Σj. We denote by T(zj) a tile centered at zj, and by Tj the set of tiles associated with the shell Sj.

Tj={T(z_j) :z_j∈Σ˜_j}. (4.3)

We chooseε₀ to satisfy two properties. First, for anyz∈ Sj, there is ˜z_j∈Σ˜_j such that

z∈ \

y∈T(˜zj)

C(y).

(4.4)

This is ensured by the choice of a small enough ε₀. Indeed, let z_j∈Σ_j be a site realizing the minimum of {kz−yk:y∈Σ_j}. There is λ >0 and u∈ B(z_j,1), such thatz=λu. Now, there is ˜z_j∈Σ˜_j such that kz˜_j−z_jk< ε₀h_j, and for anyy∈ T(˜z_j), we have ky−z_jk<2ε₀h_j. Thus, forε₀ small enough so that 1 + 2ε₀h_j≤h_j/2,

∀y∈ T(˜z_j) ku−yk ≤ ku−z_jk+kz_j−yk ≤1 + 2ε₀h_j≤hj

2 , which implies (4.4). Second, the size of a tile should be such that for some κ >1, for anyj≥1, and any tile T ∈ Tj

sup

z∈B(0,rj−hj)

P_z(S(H(Σ_j))∈ T)≤κ−1 κ . (4.5)

Inequality (4.5) follows from Lemma 5(b) of [10] (or Lemma5.1below) which for a constant J_d yields

sup

z∈B(0,rj−hj)

P_z(S(H(Σ_j))∈ T)≤J_d |T | h^d−1_j . The choice of ε₀ is such that J_d|T | ≤ ^κ−_κ¹h^d−1_j .

4.2. Bounding inner fluctuations. Forn≥0, we takeN=|B_n|, we recall thatA^∗_k(N)⊂ A^∗_k+1(N) for k∈N, and A^∗(N) =S

k≥1A^∗_k(N). We consider T^∗= min

k≥1 :[

j<k

Sj6⊂ A^∗k(N)

. (4.6)

Note that A^∗_k(N)⊂S

j<kSj, so that T^∗ is the first time k when the kth wave does not cover all its allowed space. We recall that time counts the number of exploration waves.

For the flashing process ifS

j<kSj6⊂ A^∗_k(N), then for anyk^′> k, we have S

j<kSj6⊂ A^∗_k^′(N), so that T^∗ is also the shell label where the first hole of A^∗(N) appears. We have, for l withr_l< n,

P(T^∗≤l) =P(B(0, r_l+h_l)6⊂ A^∗(N))≤X

k≤l

P(T^∗=k+ 1).

(4.7)

In this section, we estimate from above the probability P(T^∗=k+ 1) as- sumingr_k< n.

For k≥1 and Λ⊂Σ_k, we callW_k(Λ) the number of unsettled explorers that stand in Λ after thekth wave, that is,

W_k(Λ) =

N

X

i=1

1Λ(ξ_k(i)).

(4.8)

We now look at thecrossings of tiles ofTk. On the one hand, we will use that ifW_k(T) islarge, then it is unlikely that a hole appears in the cell containing T during the k+ 1th-wave. We use for this purpose the fact that covering for the flashing process is similar to filling an album for a coupon-collector model. On the other hand, ifr_k is small, it is unlikely that W_k(T) is small.

We now make precise what we intend by small and large. For any positive constant ξ, we write

P(T^∗=k+ 1) =P(T^∗=k+ 1,∀T ∈ Tk, W_k(T)≥ξ) +P(T^∗=k+ 1,∃T ∈ Tk, W_k(T)< ξ) (4.9)

≤P(T^∗=k+ 1|∀T ∈ Tk, W_k(T)≥ξ) +P(∃T ∈ Tk, W_k(T)< ξ).

A coupon-collector estimate. The first term in the right-hand side of (4.9) is bounded using a simple coupon-collector argument. Indeed, the event {T^∗=k+ 1} implies that there is an uncovered site in Sk, say z, when explorers stopped in Σ_k are released. By (4.4), there isz_k∈Σ˜_k, such that z is a possible settling position of all explorers stopped inT(z_k). Now, knowing that {W_k(T(z_k))≥ξ}, Proposition 3.1 tells us that the probability of not covering this site is less than (1−α1/h^d_k) to the power ξ. In other words,

P(T^∗=k+ 1|∀T ∈ Tk, W_k(T)≥ξ)≤ |Sk|

1−α₁ h^d_k

ξ

≤ |Sk|exp

−α₁ ξ h^d_k

.

Henceforth, we set

ξ=Ah^dlog(n) with h= sup{h_k:r_k≤n} (4.10)

and A large enough so that X

k:rk<n

P(T^∗=k+ 1|∀T ∈ Tk, W_k(T)≥ξ) (4.11)

≤ |B_n|exp(−α₁Alogn)≤ 1 n².

Estimating{W_k(T)< ξ}. For anyT ∈ Tk, we consider the counting vari- ableL_k(T) =M(B(0, r_k−h_k), r_k,T), and define

M_k(T) =W_k(T) +M(A^∗_k, r_k,T) (4.12)

so that M_k(T)^law= M(N1_{0}, r_k,T).

The idea of definingM_kandL_k(for the internal DLA process), and bounding W_k by M_k−L_k, is introduced in [10]. Our main observation is that L_k(T) is independent of W_k(T), and

W_k(T) +L_k(T)≥M_k(T).

As a consequence, for any positive constantstand ξ(and with the notation X¯ =X−E[X]),

P(W_k(T)< ξ)≤e^tξ×E[exp(−tW_k(T))] =e^tξE[exp(−t(W_k(T) +L_k(T)))]

E[exp(−tL_k(T))]

≤exp(−t(E[M_k(T)−L_k(T)]−ξ))×E[exp(−t( ¯M_k(T)))]

E[exp(−tL¯_k(T))] . Using Lemma2.3 with condition (4.5), we obtain

logP(W_k(T)< ξ)≤ −t(E[M_k(T)−L_k(T)]−ξ) +f(−t)E[M_k(T)−L_k(T)]

+κ

2g(−t) X

y∈B(0,rk−hk)

P_y²(S(H(Σ_k))∈ T).

We now proceed in two steps. We show in step 1 that for some constantκ^′, E[M_k(T)−L_k(T)]≥κ^′(n^d−(r_k−h_k)^d)h^d_k⁻¹

r_k^d−1.

Since {h_k/r_k, k≥0} is nonincreasing, it follows that there is a constant κ1 >0 such that, for all α >0, and kα := sup{j ∈N:rj < n−αhlogn}, whereh is defined in (4.10), we have

kinf≤kα

E[M_k(T)−L_k(T)]≥κ^′(n^d−(n−h)^d)h^d−1

n^d−1 ≥κ₁αh^dlogn.

Now, if we choose ξ as in (4.10), with α= 2A/κ₁ andk^∗=k_α, that is, k^∗:= sup

j∈N:r_j≤n−2A

κ₁hlog(n)

,

then, we get, for allk≤k^∗,

E[M_k(T)−L_k(T)]≥2ξ.

(4.13)

We show in step 2, that for a constant C depending on the dimension only

X

y∈B(0,rk−hk)

P²_y(S(H(Σ_k))∈ T)≤CE[M_k(T)−L_k(T)].

(4.14)

Suppose for a moment that steps 1 and 2 hold. Since, for some c >0, max(f(−t), g(−t))≤ct² when t≤1, there isc^′>0 such that for k≤k^∗

logP(W_k(T)< Ah^dlog(n))

≤ inf

0≤t≤1

−t+c

1 +Cκ 2

t²

E[M_k(T)−L_k(T)]

(4.15)

≤ −c^′E[M_k(T)−L_k(T)]≤ −2c^′Ah^dlog(n).

Now, using (4.9), (4.11) and (4.15) for Alarge enough, we have X

k<k^∗

P(T^∗=k)≤ 2 n².

Borel–Cantelli’s lemma yields then the inner control of Proposition1.5.

Step 1. We invoke Corollary5.4, withn=r_k, and ∆_n=h_k[the hypothe- sesh_k=O(r_k^1/3) andh_klarge enough hold here, as seen in the first paragraph of Section 3.1]. We have for some positive constants κ^′, K and for n large enough,

E[M_k(T)−L_k(T)] =E[M((|B_n| − |B_r

k−hk|)10, r_k,T)]

+E[M(|B_r

k−hk|10, r_k,T)]−E[M(B_r

k−hk, r_k,T)]

≥(|B_n| − |B_r

k−hk|)P₀(S(H_k)∈ T)−Kh^d_k⁻¹ (4.16)

≥2κ^′(n^d−(r_k−h_k)^d)h^d_k⁻¹

r^d_k⁻¹ −Kh^d_k⁻¹

≥κ^′(n^d−(r_k−h_k)^d)h^d_k⁻¹ r^d_k⁻¹ forr_k≤nand h₀ large enough.